Absolute Value Functions (HSC SSCE Mathematics Advanced): Revision Notes

Absolute Value Functions

What is an absolute value function?

An absolute value function is a special type of function with the general form:

$f(x) = |ax + b|$

where $a$ and $b$ are constants, and $a \neq 0$.

The absolute value operation ensures that the function's output is always non-negative. This creates a distinctive V-shaped graph that is easy to recognise and work with.

The key feature of an absolute value function is its vertex, which is the turning point where the two arms of the V meet. To find the vertex, we set the expression inside the absolute value equal to zero:

$ax + b = 0$

Solving for $x$ gives:

$x = -\frac{b}{a}$

At this $x$-value, the function output is zero, so the vertex is located at the point $\left(-\frac{b}{a}, 0\right)$.

Key features of absolute value functions

Understanding the key features helps you sketch and analyse these functions effectively.

Domain: The domain of $f(x) = |ax + b|$ is all real numbers, $\mathbb{R}$. You can substitute any value of $x$ into the function.

Range: The range is $y \geq 0$. Since the absolute value always produces non-negative outputs, the graph never goes below the $x$-axis.

Axis of symmetry: The graph has a vertical line of symmetry passing through the vertex at $x = -\frac{b}{a}$. Points on either side of this line are equidistant and have equal $y$-values.

Shape: The graph forms a V-shape, with two straight arms meeting at the vertex.

How parameters affect the graph

The constants $a$ and $b$ in $f(x) = |ax + b|$ control different aspects of the graph's appearance and position.

The parameter $a$

The parameter a controls how steep the sides of the V-shape are:

• The larger the absolute value of $a$, the steeper the arms
• If $|a| < 1$, the graph is wider and less steep
• If $a < 0$, the graph is identical to $|a| > 0$ but reflected across the $y$-axis
• The notation $-a|x|$ reflects the entire graph across the $x$-axis

The parameter $b$

The parameter b shifts the vertex horizontally:

• The vertex moves $\frac{b}{a}$ units along the $x$-axis from the origin
• If $b > 0$ in $|x - b|$, the vertex shifts right
• If $b < 0$ in $|x - b|$, the vertex shifts left

The vertex position is always at $x = -\frac{b}{a}$, which you can verify by setting the expression inside the absolute value to zero.

Graphing absolute value functions

To sketch an absolute value function, follow these steps:

1. Find the vertex by solving $ax + b = 0$
2. Plot the vertex point
3. Choose several $x$-values on either side of the vertex
4. Calculate the corresponding $y$-values
5. Plot these points and connect them to form the V-shape

Solving absolute value equations

An absolute value equation has the form:

$|ax + b| = k$

where $k$ is a non-negative constant. To solve these equations, we need to consider what values make the expression inside the absolute value equal to $k$ or $-k$.

Algebraic method:

To solve $|ax + b| = k$ algebraically:

1. Consider two cases: $ax + b = k$ and $ax + b = -k$
2. Solve each equation separately
3. Check both solutions in the original equation

Graphical method:

To solve graphically, find where the graph of $f(x) = |ax + b|$ intersects the horizontal line $y = k$.

Using technology to graph

Graphing calculators and software can help you visualise absolute value functions quickly.

Real-world application

Absolute value functions can model situations involving deviation from a target value.